Define τ⊂ P(ℝ) as follows:

τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .

prove that τ is the topology on ℝ .

Is this \(\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}

-\pi,\pi)\subseteq U\right\}\)? What's wrong? clearly \(\displaystyle \varnothing,\mathbb{R}\in T\) if \(\displaystyle (-\pi,\pi)\subseteq U_\alpha\) then \(\displaystyle (-\pi,\pi)\subseteq\bigcup_{\alpha\in\mathcal{A}}U_\alpha\) and similarly for the intersection. Or, is this \(\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:\{-\pi,\pi\}\subseteq U\right\}\)? It's the same.